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          Problem 345
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<h1 id="Problem-345"><a href="#Problem-345" class="headerlink" title="Problem 345"></a><a href="https://projecteuler.net/problem=345" target="_blank" rel="noopener">Problem 345</a></h1><hr>
<p><strong>Matrix Sum</strong></p>
<p>We define the Matrix Sum of a matrix as the maximum sum of matrix elements with each element being the only one in his row and column. For example, the Matrix Sum of the matrix below equals 3315 ( = 863 + 383 + 343 + 959 + 767):</p>
<p style="text-align:center;font-family:courier new;">
&nbsp;&nbsp;7 &nbsp;53 183 439 <span style="color:#00ff00;font-family:courier new;"><b>863</b></span>
497 <span style="color:#00ff00;font-family:courier new;"><b>383</b></span> 563 &nbsp;79 973
287 &nbsp;63 <span style="color:#00ff00;font-family:courier new;"><b>343</b></span> 169 583
627 343 773 <span style="color:#00ff00;font-family:courier new;"><b>959</b></span> 943
<span style="color:#00ff00;font-family:courier new;"><b>767</b></span> 473 103 699 303
</p>

<p>Find the Matrix Sum of:</p>
<p style="text-align:center;font-family:courier new;">
&nbsp;&nbsp;7 &nbsp;53 183 439 863 497 383 563 &nbsp;79 973 287 &nbsp;63 343 169 583
627 343 773 959 943 767 473 103 699 303 957 703 583 639 913
447 283 463 &nbsp;29 &nbsp;23 487 463 993 119 883 327 493 423 159 743
217 623 &nbsp;&nbsp;3 399 853 407 103 983 &nbsp;89 463 290 516 212 462 350
960 376 682 962 300 780 486 502 912 800 250 346 172 812 350
870 456 192 162 593 473 915 &nbsp;45 989 873 823 965 425 329 803
973 965 905 919 133 673 665 235 509 613 673 815 165 992 326
322 148 972 962 286 255 941 541 265 323 925 281 601 &nbsp;95 973
445 721 &nbsp;11 525 473 &nbsp;65 511 164 138 672 &nbsp;18 428 154 448 848
414 456 310 312 798 104 566 520 302 248 694 976 430 392 198
184 829 373 181 631 101 969 613 840 740 778 458 284 760 390
821 461 843 513 &nbsp;17 901 711 993 293 157 274 &nbsp;94 192 156 574
&nbsp;34 124 &nbsp;&nbsp;4 878 450 476 712 914 838 669 875 299 823 329 699
815 559 813 459 522 788 168 586 966 232 308 833 251 631 107
813 883 451 509 615 &nbsp;77 281 613 459 205 380 274 302 &nbsp;35 805</p>

<hr>
<p><strong>矩阵和</strong></p>
<p>在一个矩阵中选择部分元素，使得每个元素都是所在的行和列中唯一被选中的元素，这些元素的和的最大值定义为矩阵的矩阵和。例如，如下矩阵的矩阵和为3315 ( = 863 + 383 + 343 + 959 + 767)：</p>
<p style="text-align:center;font-family:courier new;">
&nbsp;&nbsp;7 &nbsp;53 183 439 <span style="color:#00ff00;font-family:courier new;"><b>863</b></span>
497 <span style="color:#00ff00;font-family:courier new;"><b>383</b></span> 563 &nbsp;79 973
287 &nbsp;63 <span style="color:#00ff00;font-family:courier new;"><b>343</b></span> 169 583
627 343 773 <span style="color:#00ff00;font-family:courier new;"><b>959</b></span> 943
<span style="color:#00ff00;font-family:courier new;"><b>767</b></span> 473 103 699 303
</p>

<p>求如下矩阵的矩阵和：</p>
<p style="text-align:center;font-family:courier new;">
&nbsp;&nbsp;7 &nbsp;53 183 439 863 497 383 563 &nbsp;79 973 287 &nbsp;63 343 169 583
627 343 773 959 943 767 473 103 699 303 957 703 583 639 913
447 283 463 &nbsp;29 &nbsp;23 487 463 993 119 883 327 493 423 159 743
217 623 &nbsp;&nbsp;3 399 853 407 103 983 &nbsp;89 463 290 516 212 462 350
960 376 682 962 300 780 486 502 912 800 250 346 172 812 350
870 456 192 162 593 473 915 &nbsp;45 989 873 823 965 425 329 803
973 965 905 919 133 673 665 235 509 613 673 815 165 992 326
322 148 972 962 286 255 941 541 265 323 925 281 601 &nbsp;95 973
445 721 &nbsp;11 525 473 &nbsp;65 511 164 138 672 &nbsp;18 428 154 448 848
414 456 310 312 798 104 566 520 302 248 694 976 430 392 198
184 829 373 181 631 101 969 613 840 740 778 458 284 760 390
821 461 843 513 &nbsp;17 901 711 993 293 157 274 &nbsp;94 192 156 574
&nbsp;34 124 &nbsp;&nbsp;4 878 450 476 712 914 838 669 875 299 823 329 699
815 559 813 459 522 788 168 586 966 232 308 833 251 631 107
813 883 451 509 615 &nbsp;77 281 613 459 205 380 274 302 &nbsp;35 805</p>

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